GLM I An Introduction to Generalized Linear Models CAS Ratemaking - PDF document

GLM I An Introduction to Generalized Linear Models CAS Ratemaking and Product Management Seminar March 2012 Presented by: Tanya D. Havlicek, ACAS, MAAA ANTITRUST Notice The Casualty Actuarial Society is committed to adhering strictly to the letter and spirit of the antitrust laws. Seminars conducted under the auspices of the CAS are designed solely to provide a forum for the expression of various points of view on topics described in the programs or agendas for such meetings. Under no circumstances shall CAS seminars be used as a means for competing companies or firms to reach any understanding – expressed or implied – that restricts competition or in any way impairs the ability of members to exercise independent business judgment regarding matters affecting competition. It is the responsibility of all seminar participants to be aware of antitrust regulations, to prevent any written or verbal discussions that appear to violate these laws, and to adhere in every respect to the CAS antitrust compliance policy. 1 Outline  Overview of Statistical Modeling  Linear Models ANOVA – Simple Linear Regression – Multiple Linear Regression – Categorical Variables – Transformations –  Generalized Linear Models Why GLM? – From Linear to GLM – Basic Components of GLM’s – Common GLM structures –  References 2

Generic Modeling Schematic Predictor Vars Response Vars Weights Driver Age Losses Claims Region Default Exposures Relative Equity Persistency Premium Credit Score Statistical Model Model Results Parameters Validation Statistics 3 Basic Linear Model Structures - Overview  Simple ANOVA : – Y ij = µ + e ij or more generally Y ij = µ + ψ i + e ij – In Words: Y is equal to the mean for the group with random variation and possibly fixed variation – Traditional Classification Rating – Group Means – Assumptions: errors independent & follow N(0, σ e2 ) – ∑ ψ i = 0 i = 1,…..,k (fixed effects model) – ψ i ~ N(0, σ ψ 2 ) (random effects model) 4 Basic Linear Model Structures - Overview  Simple Linear Regression : y i = b o + b 1 x i + e i – Assumptions: • linear relationship • errors independent and follow N(0, σ e2 )  Multiple Regression : y i = b o + b 1 x 1i + ….+ b n x ni + e i Assumptions: same, but with n independent random variables (RV’s) –  Transformed Regression : transform x, y, or both; maintain errors are N(0, σ e2 ) y i = exp(x i )  log(y i ) = x i 5

Simple Regression (special case of multiple regression)  Model: Y i = b o + b 1 X i + e i Y is the dependent variable explained by X, the – independent variable Y could be Pure Premium, Default Frequency, etc – Want to estimate relationship of how Y depends on X – using observed data Prediction: Y= b o + b 1 x* for some new x* (usually – with some confidence interval) 6 Simple Regression – A formalization of best fitting a line through data with a ruler and a pencil N – Correlative relationship    ( Y Y )( X X ) i i       – Simple e.g. determine a trend to apply i 1 , a Y X N   2 ( X X ) i  i 1 Mortgage Insurance Average Claim Paid Trend 70,000 60,000 50,000 Severity 40,000 Severity Predicted Y 30,000 20,000 10,000 0 1985 1990 1995 2000 2005 2010 Accident Year 7 Note: All data in this presentation are for illustrative purposes only Regression – Observe Data 8

Regression – Observe Data Foreclosure Hazard vs Borrower Equity Position 8 7 Relative Foreclosure Hazard 6 5 4 3 2 1 0 -50 -25 0 25 50 75 100 125 Equity as % of Original Mortgage 9 Regression – Observe Data Foreclosure Hazard vs Borrower Equity Position <20% 8 7 Relative Foreclosure Hazard 6 5 4 3 2 1 0 -50 -40 -30 -20 -10 0 10 20 Equity as % of Original Mortgage 10 Simple Regression ANOVA Significance F df SS MS F Regression 1 52.7482 52.7482 848.2740 <0.0001 Residual 17 1.0571 0.0622 Total 18 53.8053  How much of the sum of squares is explained by the regression? SS = Sum Squared Errors SSTotal = SSRegression + SSResidual (Residual also called Error) SSTotal = ∑ ( y i – y ) 2 = 53.8053 SSRegression = b 1 est *[ ∑ x i y i -1/n( ∑ x i )( ∑ y i )] = 52.7482 SSResidual = ∑ (y i – y i est ) 2 = SSTotal – SSRegression 1.0571 = 53.8053 – 52.742 11

Simple Regression ANOVA df SS MS F Significance F Regression 1 52.7482 52.7482 848.2740 <0.0001 Residual 17 1.0571 0.0622 Total 18 53.8053 Regression Statistics Multiple R 0.9901  MS = SS divided by df R Square 0.9804  R 2 : (SS Regression/SS Total) Adjusted R Square 0.9792 0.9804 = 52.7482 / 53.8053 percent of variance explained –  F statistic: (MS Regression/MS Residual)  significance of regression: F tests H o : b 1 =0 v. H A : b 1 ≠ 0 – 12 Simple Regression Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 3.3630 0.0730 46.0615 0.0000 3.2090 3.5170 3.2090 3.5170 X -0.0828 0.0028 -29.1251 0.0000 -0.0888 -0.0768 -0.0888 -0.0768 T statistics: (b i est – H o (b i )) / s.e.(b i est ) • significance of individual coefficients • T 2 = F for b 1 in simple regression • (-29.1251) 2 = 848.2740 • F in multiple regression tests that at least one coefficient is nonzero. For the simple case, at least one is the same as the entire model. F stat tests the global null model. 13 Residuals Plot  Looks at (y obs – y pred ) vs. y pred  Can assess linearity assumption, constant variance of errors, and look for outliers  Standardized Residuals (raw residual scaled by standard error) should be random scatter around 0, standard residuals should lie between -2 and 2  With small data sets, it can be difficult to assess assumptions Plot of Standardized Residuals 2 1.5 1 Standardized Residual 0.5 0 -0.5 -1 -1.5 -2 -2.5 0 1 2 3 4 5 6 7 8 Predicted Foreclosure Hazard 14

Normal Probability Plot  Can evaluate assumption e i ~ N(0, σ e2 ) Plot should be a straight line with intercept µ and slope σ e2 – Can be difficult to assess with small sample sizes – Normal Probability Plot of Residuals Standard Residuals 4 3.5 3 Standardized Residual 2.5 2 1.5 1 0.5 0 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 -0.5 -1 -1.5 Theoretical z Percentile 15 Residuals  If absolute size of residuals increases as predicted value increases, may indicate nonconstant variance  May indicate need to transform dependent variable  May need to use weighted regression  May indicate a nonlinear relationship Plot of Standardized Residuals Standard Residuals 3 2 Standardized Residual 1 0 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 -1 -2 -3 Predicted Severity 16 Distribution of Observations  Average claim amounts for Rural drivers are normally distributed as are average claim amounts for Urban drivers  Mean for Urban drivers is twice that of Rural drivers  The variance of the observations is equal for Rural and Urban  The total distribution of average claim amounts across Rural and Urban is not Normal here it is bimodal – Distribution of Individual Observations Rural Urban µ R µ U 17

Distribution of Observations  The basic form of the regression model is Y = b o + b 1 X + e  µ i = E[Y i ] = E[b o + b 1 X i + e i ] = b o + b 1 X i + E[e i ] = b o + b 1 X i  The mean value of Y, rather than Y itself, is a linear function of X  The observations Y i are normally distributed about their mean µ i Y i ~ N( µ i , σ e2 ) Each Y i can have a different mean µ i but the variance σ e2 is the same for each  observation Y Line Y = b o + b 1 X b o + b 1 X 2 b o + b 1 X 1 X X 2 X 1 18 Multiple Regression (special case of a GLM)  Y = β 0 + β 1 X 1 + β 2 X 2 + … + β n X n + ε  E[Y] = β X β is a vector of the parameter coefficients Y is a vector of the dependent variable X is a matrix of the independent variables – Each column is a variable – Each row is an observation  Same assumptions as simple regression 1) model is correct (there exists a linear relationship) 2) errors are independent 3) variance of e i constant 4) e i ~ N(0, σ e2 )  Added assumption the n variables are independent 19 Multiple Regression  Uses more than one variable in regression model – R-sq always goes up as add variables – Adjusted R-Square puts models on more equal footing – Many variables may be insignificant  Approaches to model building – Forward Selection - Add in variables, keep if “significant” – Backward Elimination - Start with all variables, remove if not “significant” – Fully Stepwise Procedures – Combination of Forward and Backward 20